Mirrors and Plane Refraction — Part 1: Reflection at spherical mirrors: visual lesson, ISC Class 12 Physics

Lesson

Reflection at Spherical Mirrors

Part 1 — curved mirrors: how they form images, the sign convention and the mirror formula.

In this lesson you will

Use the terms pole, focus, centre and radius of curvature, and show that R = 2f for paraxial rays
Apply the New Cartesian sign convention and derive the mirror formula for a concave mirror
Find the position, size and nature of images in concave and convex mirrors

Concept

The parts of a spherical mirror

A spherical mirror is part of a hollow sphere: concave if the inside is silvered, convex if the outside is.

Pole P: centre of the mirror’s surface. Centre of curvature C: centre of the sphere. Radius of curvature R: PC.
Principal axis: the line through P and C. Aperture: the width of the mirror.
Principal focus F: where rays parallel to the axis meet after reflection (concave), or seem to come from (convex). Focal length f: PF.
Paraxial rays: rays close and nearly parallel to the axis; all the results here assume them.

Diagram

Why the focal length is half the radius

A ray parallel to the axis strikes the mirror at M; CM is the normal. Angle of incidence $\theta$ = angle of reflection $\theta$ , and $\angle MCF = \theta$ (alternate angles). So triangle FCM is isosceles: $FC = FM$ . For a paraxial ray M is very close to P, so $FM \approx FP$ . Then $FC = FP$ : F is the midpoint of PC. $f = \dfrac{R}{2}$

The same proof works for a convex mirror, with F behind it.
For wide rays F moves towards the mirror (spherical aberration), so only paraxial rays meet at one point.

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